Dirichlet series
| L(s) = 1 | − 17·2-s + 324·3-s − 405·4-s + 2.50e3·5-s − 5.50e3·6-s − 9.60e3·7-s + 4.53e3·8-s + 6.56e4·9-s − 4.25e4·10-s + 1.63e4·11-s − 1.31e5·12-s − 8.49e4·13-s + 1.63e5·14-s + 8.10e5·15-s − 1.85e5·16-s − 4.51e5·17-s − 1.11e6·18-s − 1.03e6·19-s − 1.01e6·20-s − 3.11e6·21-s − 2.78e5·22-s − 5.45e6·23-s + 1.46e6·24-s + 3.90e6·25-s + 1.44e6·26-s + 1.06e7·27-s + 3.88e6·28-s + ⋯ |
| L(s) = 1 | − 0.751·2-s + 2.30·3-s − 0.791·4-s + 1.78·5-s − 1.73·6-s − 1.51·7-s + 0.391·8-s + 10/3·9-s − 1.34·10-s + 0.337·11-s − 1.82·12-s − 0.824·13-s + 1.13·14-s + 4.13·15-s − 0.709·16-s − 1.31·17-s − 2.50·18-s − 1.81·19-s − 1.41·20-s − 3.49·21-s − 0.253·22-s − 4.06·23-s + 0.903·24-s + 2·25-s + 0.619·26-s + 3.84·27-s + 1.19·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(3^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.55277\times 10^{6}\) |
| Root analytic conductor: | \(7.35382\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 17 T + 347 p T^{2} + 1769 p^{3} T^{3} + 2463 p^{8} T^{4} + 1769 p^{12} T^{5} + 347 p^{19} T^{6} + 17 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 16382 T + 5020122196 T^{2} - 217584776964382 T^{3} + 11921686074265839174 T^{4} - 217584776964382 p^{9} T^{5} + 5020122196 p^{18} T^{6} - 16382 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 84914 T + 2822470056 T^{2} + 512057500862326 T^{3} + 31041290629938587678 T^{4} + 512057500862326 p^{9} T^{5} + 2822470056 p^{18} T^{6} + 84914 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 451528 T + 153229030780 T^{2} + 76607762314998392 T^{3} + \)\(24\!\cdots\!18\)\( p T^{4} + 76607762314998392 p^{9} T^{5} + 153229030780 p^{18} T^{6} + 451528 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 1031478 T + 1319463192228 T^{2} + 884484808021561830 T^{3} + \)\(63\!\cdots\!26\)\( T^{4} + 884484808021561830 p^{9} T^{5} + 1319463192228 p^{18} T^{6} + 1031478 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 5459068 T + 17377312279708 T^{2} + 36490666611429158252 T^{3} + \)\(57\!\cdots\!70\)\( T^{4} + 36490666611429158252 p^{9} T^{5} + 17377312279708 p^{18} T^{6} + 5459068 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 4867288 T + 38875560332780 T^{2} + \)\(16\!\cdots\!08\)\( T^{3} + \)\(77\!\cdots\!74\)\( T^{4} + \)\(16\!\cdots\!08\)\( p^{9} T^{5} + 38875560332780 p^{18} T^{6} + 4867288 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 1098642 T + 89804308891180 T^{2} - 73998842828908472586 T^{3} + \)\(34\!\cdots\!78\)\( T^{4} - 73998842828908472586 p^{9} T^{5} + 89804308891180 p^{18} T^{6} - 1098642 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 2110068 T + 191071215259780 T^{2} + \)\(62\!\cdots\!68\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} + \)\(62\!\cdots\!68\)\( p^{9} T^{5} + 191071215259780 p^{18} T^{6} - 2110068 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 1104700 T + 39435529285876 T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(88\!\cdots\!86\)\( T^{4} - \)\(37\!\cdots\!00\)\( p^{9} T^{5} + 39435529285876 p^{18} T^{6} - 1104700 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 15322648 T + 17550681453636 p T^{2} + \)\(89\!\cdots\!32\)\( T^{3} + \)\(23\!\cdots\!90\)\( T^{4} + \)\(89\!\cdots\!32\)\( p^{9} T^{5} + 17550681453636 p^{19} T^{6} + 15322648 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 5033968 T + 4150036490198204 T^{2} + \)\(14\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!14\)\( p T^{4} + \)\(14\!\cdots\!48\)\( p^{9} T^{5} + 4150036490198204 p^{18} T^{6} + 5033968 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 149234422 T + 20731466508128464 T^{2} + \)\(16\!\cdots\!54\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} + \)\(16\!\cdots\!54\)\( p^{9} T^{5} + 20731466508128464 p^{18} T^{6} + 149234422 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 141913876 T + 17679422039240540 T^{2} + \)\(16\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!92\)\( p^{9} T^{5} + 17679422039240540 p^{18} T^{6} + 141913876 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 235578792 T + 56125259764358188 T^{2} - \)\(78\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(78\!\cdots\!84\)\( p^{9} T^{5} + 56125259764358188 p^{18} T^{6} - 235578792 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 401097064 T + 150493806142350924 T^{2} + \)\(32\!\cdots\!08\)\( T^{3} + \)\(66\!\cdots\!74\)\( T^{4} + \)\(32\!\cdots\!08\)\( p^{9} T^{5} + 150493806142350924 p^{18} T^{6} + 401097064 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 126532750 T - 32633729357185636 T^{2} + \)\(21\!\cdots\!50\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} + \)\(21\!\cdots\!50\)\( p^{9} T^{5} - 32633729357185636 p^{18} T^{6} + 126532750 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 653180926 T + 324690853793630280 T^{2} + \)\(11\!\cdots\!66\)\( T^{3} + \)\(30\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!66\)\( p^{9} T^{5} + 324690853793630280 p^{18} T^{6} + 653180926 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 624551664 T + 546300715881429372 T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(20\!\cdots\!52\)\( p^{9} T^{5} + 546300715881429372 p^{18} T^{6} + 624551664 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 702347048 T + 683213635075197836 T^{2} + \)\(36\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!78\)\( p T^{4} + \)\(36\!\cdots\!84\)\( p^{9} T^{5} + 683213635075197836 p^{18} T^{6} + 702347048 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 1510406712 T + 1090068108195555388 T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!86\)\( T^{4} + \)\(40\!\cdots\!20\)\( p^{9} T^{5} + 1090068108195555388 p^{18} T^{6} + 1510406712 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 717560562 T - 208570240917207360 T^{2} + \)\(57\!\cdots\!98\)\( T^{3} + \)\(82\!\cdots\!46\)\( T^{4} + \)\(57\!\cdots\!98\)\( p^{9} T^{5} - 208570240917207360 p^{18} T^{6} + 717560562 p^{27} T^{7} + p^{36} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.093926867785016894113332418680, −8.464160213609031671488348671228, −8.443058484159564565020031949666, −8.327119896512143481650324973893, −8.039778843695294260515015563065, −7.43494061531291443423097206917, −7.22267207747354207512602826741, −6.70967712649244316230394030096, −6.64937991337605164256892243442, −6.36720439823026026414345619419, −5.97625638898797218829355390251, −5.67132823819510780524763608657, −5.47170218012751306194882527201, −4.45915950069270873817313340279, −4.31321739449842320289732664610, −4.24629329096488217319704147542, −4.21933165283894135543247849645, −3.33572264640978475038721517661, −3.13968951796888087853293762949, −2.67305927162513893131586325845, −2.44178785649484179245220639638, −2.18665304164068342694308444292, −1.88055763959339502429222857434, −1.44307484105465897886304925768, −1.39840709862675094858904387483, 0, 0, 0, 0, 1.39840709862675094858904387483, 1.44307484105465897886304925768, 1.88055763959339502429222857434, 2.18665304164068342694308444292, 2.44178785649484179245220639638, 2.67305927162513893131586325845, 3.13968951796888087853293762949, 3.33572264640978475038721517661, 4.21933165283894135543247849645, 4.24629329096488217319704147542, 4.31321739449842320289732664610, 4.45915950069270873817313340279, 5.47170218012751306194882527201, 5.67132823819510780524763608657, 5.97625638898797218829355390251, 6.36720439823026026414345619419, 6.64937991337605164256892243442, 6.70967712649244316230394030096, 7.22267207747354207512602826741, 7.43494061531291443423097206917, 8.039778843695294260515015563065, 8.327119896512143481650324973893, 8.443058484159564565020031949666, 8.464160213609031671488348671228, 9.093926867785016894113332418680